Part 2 : Nanostuctures Module 3 : Electron Transport
Lecture 1 : Boltzmann Transport Equation (BTE)
 
This leads to
$\displaystyle {\bf {j}}\,\equiv\, \sigma\, {\bf {E}}= n \,q\,\mu\,{\bf {E}},$
 (6)
where $ \sigma$ is the conductivity defined
$\displaystyle \sigma = \frac{n\,q^2}{m}\,<\tau_m>.$
 (7)
This formula is known as Drude-conductivity formula. In semiconductors, $ m$ is replaced by the effective mass $ m^*$ of the carrier. Clearly, the friction term leads to Ohm's Law, i.e., $ \sigma$ is independent of the electric field strength.
In elementary derivations of Drude formula found in introductory text books of Solid State Physics, one uses the concept of collision time $ \tau_c$. As the carriers drift through the solid, they occasionally collide, for instance, with the vibrating atoms/ions in the lattice, and one would anticipate the existence of a mean free time between the collisions or scattering. The average time interval between successive collisions is the collision time $ \tau_c$, which is of the order of $ <\tau_m>$.
Note that $ 1/\sigma\,\equiv\,\rho$ is the resistivity (with unit $ \Omega$ cm = Volt cm/Amp) so that the mobility $ \mu$ has the unit of cm2/(Volt sec). To get an idea about the order of magnitude for $ \mu$, note that $ e/m_e$= 1.76$ \times$1015cm2/Volt and taking $ <\tau_m>\sim 10^{-13}$ sec (inverse of a typical lattice vibrational frequency), one gets $ \mu \sim$ 176 cm2/(Volt sec). In $ n$-type Germanium (Ge), $ m^*\, \sim\, 0.1m_e$, and $ <\tau_m>\sim 10^{-13}$ sec at room temperature, so that $ \mu$ for this material at room temperature is $ \mu \sim$ 1760 cm2/(Volt sec), which agrees in order of magnitude with the observed mobility of 3900 cm2/(Volt sec).